Problem: Simplify; express your answer in exponential form. Assume $a\neq 0, z\neq 0$. $\dfrac{{(a^{-2})^{2}}}{{(a^{2}z^{-3})^{3}}}$
To start, try working on the numerator and the denominator independently. In the numerator, we have ${a^{-2}}$ to the exponent ${2}$ . Now ${-2 \times 2 = -4}$ , so ${(a^{-2})^{2} = a^{-4}}$ In the denominator, we can use the distributive property of exponents. ${(a^{2}z^{-3})^{3} = (a^{2})^{3}(z^{-3})^{3}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(a^{-2})^{2}}}{{(a^{2}z^{-3})^{3}}} = \dfrac{{a^{-4}}}{{a^{6}z^{-9}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{-4}}}{{a^{6}z^{-9}}} = \dfrac{{a^{-4}}}{{a^{6}}} \cdot \dfrac{{1}}{{z^{-9}}} = a^{{-4} - {6}} \cdot z^{- {(-9)}} = a^{-10}z^{9}$.